Nonpositive Curvature: A Geometric Approach to Hilbert-Schmidt Operators

Abstract

We give a Riemannian structure to the set Σ of positive invertible unitized Hilbert–Schmidt operators, by means of the trace inner product. This metric makes of Σ a nonpositively curved, simply connected and metrically complete Hilbert manifold. The manifold Σ is a universal model for symmetric spaces of the noncompact type: any such space can be isometrically embedded into Σ. We give an intrinsic algebraic characterization of convex closed submanifolds M. We study the group of isometries of such submanifolds: we prove that GM, the Banach–Lie group generated by M, acts isometrically and transitively on M. Moreover, GM admits a polar decomposition relative to M, namely GM M × K as Hilbert manifolds (here K is the isotropy of p = 1 for the action Ig :p → gpg∗), and also GM/K M so M is an homogeneous space. We obtain several decomposition theorems by means of geodesically convex submanifolds M. These decompositions are obtained via a nonlinear but analytic orthogonal projection ΠM :Σ → M, a map which is a contraction for the geodesic distance. As a byproduct, we prove the isomorphism NM Σ (here NM stands for the normal bundle of a convex closed submanifold M). Writing down the factorizations for fixed ea, we obtain ea = ex evex with ex ∈ M and v orthogonal to M at p = 1. As a corollary we obtain decompositions for the full group of invertible elements G M × exp(T1M⊥) × K.